Using calculus: If you invest $5000 compounded continuously at 4% p.a. how much will this investment be worth in 5 years? I am not sure what e is. Should my equation be this: A(t)=Pe^rt
A(t)=(5000e^(0.04)(5))
Thanks.
e is the root of natural logarithms, 2.71828.. It has appeared in some of your previous questions, so you should be familiar with it.
e is also the limit as n-> infinity of
(1 + 1/n)^n. That is why it appears on the formula for continuous compounding.
A hand calculator will let you compute 5000^*e^0.2 easily
My other questions I didn't understand what e was either. My calculator does not allow me to perform the function that you told me. Please further explain.
Nvm, I figured this out myself. :)
Great question! To determine the value of an investment compounded continuously, we can use the formula for continuous compound interest:
A(t) = P * e^(rt),
where:
A(t) is the final amount of the investment after time t,
P is the initial principal (in this case, $5000),
e is the base of the natural logarithm (approximately 2.71828),
r is the annual interest rate (in decimal form),
and t is the time in years (in this case, 5 years).
So, your equation should indeed be:
A(t) = 5000 * e^(0.04 * 5).
To calculate the value, you can first determine the exponential term, e^(rt), and then multiply it by the principal amount:
A(t) = 5000 * e^(0.2).
Using a scientific calculator, simply plug in the value 0.2 for the exponent and then multiply it by 5000:
A(t) β 5000 * 1.22140 β $6107.
Therefore, your investment will be worth approximately $6107 after 5 years with continuous compounding at a 4% annual interest rate.
Remember that the value of 'e' is a mathematical constant and stands for the base of the natural logarithm. It appears in many areas of mathematics and is approximately equal to 2.71828.